Brownan moton Let X ={X t : t R + } be a real-valued stochastc process: a famlty of real random varables all defned on the same probablty space. Defne F t = nformaton avalable by observng the process up to tme t = what we learn by observng X s for s t Call X a standard Brownan moton f ) X, ω) s a contnuous functon on R +, for each fxed ω ) X,ω)= for all ω ) for each s, {X t X s : t s} s ndependent of F s v) X t X s s N, t s) dstrbuted for each s < t Another way to express ): ) X t X s F s N, t s) for s < t. Equvalent way to express ) and v): for each t 1 t 2... t k, the random vector X t1,...,x tk ) has a multvarate normal dstrbuton wth zero means and covarances gven by covx s, X t ) = mns, t) Useful facts. For a fxed τ defne Z t = X τ+t X τ for t Markov property: Z s a Brownan moton ndependent of F τ = nformaton avalable up to tme τ. Strong Markov property: Same asserton holds for stoppng tmes τ. Tme reversal: moton. Defne Z t = tx 1/t for t >, wth Z =. Then {Z t : t R + } s a also a Brownan Martngale propertes: Abbrevate E... F t )to E t...) The Brownan moton process s a martngale: for s < t, E s X t ) = E s X s ) + E s X t X s ) = X s The process M t = Xt 2 t s a martngale: for s < t, E s M t ) = E s Xs + X ) 2 t by ). where X := Xt X s = Xs 2 + 2X se s X) + E s X) 2 t = M s because E s X) = ande s X) 2 = t s. For each real θ, the process Y t = exp θ X t 1 2 θ 2 t ) s a martngale: for s < t, E s Y t ) = E s Ys expθ X 1 2 θ 2 t s) ) = Y s E s e θ X exp 1 2 θ 2 t s) ) = Y s because X F s N, t s). c Davd Pollard, 24 Statstcs 251/551
Lévy s martngale characterzaton of Brownan moton. Suppose {X t : t 1} a martngale wth contnuous sample paths and X =. Suppose also that X 2 t t s a martngale. Then X s a Brownan moton. Heurstcs. I ll gve a rough proof for why X 1 s N, 1) dstrbuted. Let f x, t) be a smooth functon of two arguments, x R and t [, 1]. Defne f x = f x and f xx = 2 f 2 x and f t = f t. Let h = 1/n for some large postve nteger n. Defne t = h for =, 1,...,n. Wrte X for X t + h) X t ). Then E f X 1, 1) E f X, ) = E f Xt +h, t + h) E f X t, t ) ) <n E X) f x X t, t ) + 1 2 X) 2 f xx X t, t ) + hf t X t, t ) ) <n Independence of X from F t gves a factorzaton for the th sumand: E X)E f x X t, t ) + 1 2 E X) 2 E f xx X t, t ) = + 1 2 he f xxx t, t ) The sum then takes the form of an approxatng sum for the ntegral 1 2 E f xxx s, s) + E f t X s, s) ) ds If we pad more attenton to the errors of approxmaton we would see that ther contrbutons go to zero as the {t } grd gets fner. In the lmt we have t E f X 1, 1) E f X, ) = E 1 2 f xxx s, s) + f t X s, s) ) ds Now specalze to the case f x, s) = exp θ x 1 2 θ 2 s ), wth θ a fxed real constant. By drect calculaton, we have f x = θ f x, s) and f xx = θ 2 f x, s) and f t = 1 2 θ 2 f x, s) Thus Ee θ X 1 e θ 2 /2 1 = ds =. That s, X 1 has the moment generatng functon expθ 2 /2), whch dentfes t as havng a N, 1) dstrbuton. How to buld a Brownan moton In Eucldean space wth e 1,...,e N orthogonal unt vectors, f z span{e 1,...,e N } then z = α N e wth α = z, e nner product) For real-valued functons f and g on, 1] defne f, g = f x)gx) dx and f = f, f. Let e 1,...,e N be real-valued functons on, 1] wth { 1 f = j e, e j := f j c Davd Pollard, 24 Statstcs 251/551
If f, g E N := span{e 1,...,e N } then f x) = α N e x) wth α = f, e gx) = β N e x) wth β = g, e Consequence: f, g = α, j β j e, e j = α N β = f, e N g, e Random coeffcents. Wrte L 2 for the set of functons f on, 1] wth f x)2 dx <. Let η 1,η 2,... be ndependent, each N, 1) dstrbuted. For each f n L 2 defne Z N f ) := f, e N η The random varables {Z N f ) : f L 2 } have a jont normal dstrbuton wth zero means and covarances gven by covz N f ), Z N g)) =, j f, e covη,η j ) g, e j = N f, e g, e In partcular, f f, g E N then covz N f ), Z N g)) = f, g and Z N f ) N, f 2 ). { 1 For t 1 defne f t x) = f x t otherwse. Then f s, f t = 1{x s} 1{x t} dx = mns, t). If f t E N then Z N f t ) N, t) If f t, f s E N then covz N f s ), Z N f t )) = mns, t). Let N tend to. Convergence? In the lmt f t exsts) we have a Gaussan stochastc process {Z f t ) : t 1} wth the means and varances desred for Brownan moton. Contnuty of sample paths. For k =, 1,... and < 2 k defne functons on, 1] by H,k x) = 1{2 k < x + 1 / 2 )2 k } 1{ + 1 / 2 )2 k < x + 1)2 k } Note that H,k s the ndcator functon of the nterval J,k = /2 k, + 1)/2 k ] and H,k x) 2 dx = {x J,k } dx = 2 k. Haar bass) The functons e,k x) = 2 k/2 H,k x) 2 satsfy { e,k, e,k = 1 f = and k = k otherwse Moreover, each e,k s orthogonal to the constant functon Ux) 1, n the sense that e,k, U =. Redefne E N to be E N = span {U} {e,k : < 2 k, k =, 1,...,N} ) It s not hard to show that E N conssts of all those real-valued functons on, 1] that take a constant value on each J,k subnterval. Let η and η,k for < 2 k and k =, 1,... be ndependent N, 1) random varables. Usng the Haar bass functons we have, for t 1, Z N f t ) = f t, U η + N f k= <2 t, e k,k η,k = tη + N <1> k= 2k/2 X k t) where X k t) = f <2 t, H k,k η,k c Davd Pollard, 24 Statstcs 251/551
As a functon of t, each f t, H,k s nonzero only n the nterval J,k, wthn whch t s pecewse lnear, achevng ts maxmum value of 2 k+1) at the mdpont, 2 + 1)/2 k+1 : f t,h,k 2 k+1) 2/2 k+1 2+1)/2 k+1 2+2)/2 k+1 The process X k t) has contnuous, pecewse lnear sample paths. It takes the value at t = /2 k for =, 1,...,2 k. It takes the value η,k /2 k+1 at the pont 2 + 1)/2 k+1. X 2 t) η 1,2 η 3,2 1 η,2 η 2,2 Thus max t X k t) =2 k+1) max η,k. From HW sheet 5, E max X k t) =2 k+1) E max η,k 2 k+1) 2log2 k ) t and hence E k= 2k/2 max t X k t) k= 2k/2 2 k+1) 2log2 k )<. The random varable k= 2k/2 max t X k t) has a fnte expectaton. It must be fnte everywhere, except possbly on a set c wth zero probablty. For all sample ponts ω n, the sum n <1> converges unformly n t. The sample paths of the lmt process, beng unform lmts of functons contnuous n t, are contnuous, at least for ω n. Fx sample paths for ω c? Total varaton and quadratc varaton of a functon Let f be a real-valued functon on [, 1]. Defne the total varaton of f by V f ) = sup f t +1) f t ), G where G ranges over all fnte grds = t < t 1,... < t k = 1. Say that f s of bounded varaton f V f )<. Defne quadratc varaton of f for a grd G as Q f, G) = f t +1) f t ) 2 where G s the grd = t < t 1,...<t k = 1. Fact: If f s contnuous and has bounded varaton then Q f, G) asmeshg), where meshg) = max t +1 t. Proof. Gven ɛ>, there exsts a δ>such that f s) f t) ɛ whenever s t δ. The functon f s also unformly contnuous on [, 1]. ) For any grd G wth meshg) δ we have Q f, G) ɛ f t +1) f t ) ɛv f ) And so on. c Davd Pollard, 24 Statstcs 251/551
Total varaton and quadratc varaton of Brownan moton sample paths Let {X t : t 1} be the ntal chunk of a standard Brownan moton. Fact: Almost almost all sample paths of X have nfnte total varaton. Proof. Wrte V ω) for the total varaton of the sample path X,ω). For each postve nteger n, defne V n ω) = X t +1,ω) X t,ω) where t = /n for =, 1,...,n. The absolute value of the ncrements X = X t +1,ω) X t,ω) are ndependent wth mean c / n and varance less than c 1 /n, for some constants c and c 1. Thus EV n = c n and varvn ) c 1. By Tchebychev s nequalty, P{V n 1 2 c n}=p{vn c n 1 2 c varv n ) n} 1 1 2 c 1 n) 2 Complete the proof by notng that V ω) V n ω) for every n. as n. Fact: For any sequence of grds G n wth meshg n ), QX,ω),G n ) 1 n probablty Proof. Abbrevate QX,ω),G n ) to Q n ω). That s, Q n ω) = X) 2 where δ X are the ncrements n X for grd G n. Notce that EQ n = E X) 2 = ) t+1 t = 1 and, by the ndependence of the ncrements, varq n ) = var X) 2) E X) 4 ) 2 c 2 t+1 t c2 meshg n ), for some constant c 2. c Davd Pollard, 24 Statstcs 251/551